What a Derivative Means in a Population Context
Suppose represents the size of a population at time . The derivative tells you how quickly that population is changing at a particular instant.
The key idea is simple: the derivative of a population function is the instantaneous rate of population change.
If is measured in people and is measured in years, then is measured in people per year.
If , then the population is increasing at that instant. If , the population is decreasing. If , the population is momentarily constant.
Population vs. Population Growth Rate
One of the most important distinctions on the AP Calculus exam is the difference between the population itself and its rate of change.
Population:
Population growth rate:
Population acceleration:
For example, a population of 50,000 people and a population growth rate of 2,000 people per year are describing completely different quantities. The first is the population size; the second describes how rapidly that size is changing.
Positive and Negative Growth
The sign of the derivative tells you whether the population is increasing or decreasing.
A population is modeled by , where is measured in years and is measured in people.
Births, Deaths, and Net Population Change
Population change can often be understood as the difference between the rate at which individuals enter a population and the rate at which individuals leave it.
Here, represents the birth or immigration rate, while represents the death or emigration rate.
This gives a useful way to interpret a derivative in context. If a population has 4,500 births per year and 3,200 deaths per year, its net growth rate is 1,300 people per year.
A city's population is changing according to the model
where is the annual birth rate and is the annual death rate.
Finding the Instantaneous Growth Rate from a Population Model
Many AP Calculus problems give you an explicit formula for a population and ask for its rate of change at a particular time. The process is always the same:
- Identify the population function.
- Differentiate the function.
- Evaluate the derivative at the requested time.
- Include appropriate units and interpret the result.
A population is modeled by
where is measured in years and is measured in thousands of people.
Average Rate of Growth vs. Instantaneous Rate
Do not confuse the derivative with an average rate of change over an interval.
The average rate describes what happened across an interval. The derivative describes what is happening at one particular instant.
A population changes from 80,000 people at to 92,000 people at .
That does not mean the population was increasing at exactly 3,000 people per year at every instant. The instantaneous rate could have been different at different times.
Relative Growth Rate
Sometimes the raw growth rate is not the most useful way to compare populations. A population of 10,000 growing by 500 people per year is changing much more rapidly, relative to its size, than a population of 1,000,000 growing by the same 500 people per year.
The relative growth rate compares the instantaneous growth rate with the population itself.
If is measured in people and is measured in years, the relative growth rate has units of inverse years. It is often expressed as a percentage by multiplying by 100.
A population has size at a particular time, and its instantaneous growth rate is .
When Is the Population Growing the Fastest?
A population can be increasing while its growth rate is decreasing. This is one of the most important ideas to understand.
If , then the population is increasing. But if , the growth rate itself is decreasing.
So:
- : population increasing.
- : population decreasing.
- : growth rate increasing.
- : growth rate decreasing.
Suppose a population model satisfies
at a particular time.
Therefore, the population is still increasing, but it is increasing more slowly.
Logistic Population Growth
Not every population can grow exponentially forever. Resources such as food, water, space, and habitat impose limits. A common model for this situation is the logistic growth model.
Here, represents the carrying capacity of the environment, while controls the growth rate.
The derivative of a logistic population model has an especially useful interpretation.
The expression shows that population growth depends on both the current population and the amount of room remaining before the population reaches its carrying capacity.
For a logistic model, the population growth rate is greatest when the population is approximately , or one-half of the carrying capacity. At that point, the population is transitioning from accelerating growth to decelerating growth.
Interpreting a Logistic Model
A population follows a logistic model with carrying capacity .
Using a Table of Population Values
AP Calculus questions do not always give you an explicit population formula. You may instead receive a table of population values or growth rates.
When a table provides , you can use those values directly to answer questions about the population's instantaneous or approximate rate of change.
Suppose a table gives the following information:
The values represent population size in thousands.
This is a common AP Calculus strategy: when the derivative is already provided in a table, do not differentiate again. Interpret the derivative that the problem gives you.
Units Matter
Population-growth problems are contextual problems, so units are part of the mathematical answer.
- If is people and is years, then is people per year.
- If population is measured in thousands of people, the derivative is measured in thousands of people per unit of time.
- If time is measured in months, the derivative is measured in people per month.
A numerical derivative is not enough on a contextual AP Calculus question. If you calculate , your final response should say what that number represents, such as "2,400 people per year."
Common Mistakes
The population tells you how many individuals there are. The derivative tells you how quickly that number is changing.
A positive first derivative means the population is increasing. To determine whether the growth rate itself is increasing or decreasing, you need to examine the second derivative.
"500" is not a complete contextual answer. If the derivative is 500 and time is measured in years, the interpretation is 500 people per year.
The average rate of change describes an entire interval. The derivative describes the rate at one specific instant.
Real populations may experience limited resources. Logistic models account for a carrying capacity and eventually produce a decreasing growth rate as the population approaches that capacity.
Practice Problems
A population is modeled by , where is measured in years. Find the instantaneous population growth rate at .
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A town has a birth rate of 1,850 people per year and a death rate of 1,200 people per year. What is the town's net population growth rate?
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A population has size 80,000 and an instantaneous growth rate of 1,600 people per year. What is the relative growth rate as a percentage?
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A population satisfies and at time . Is the population increasing or decreasing? Is its growth rate increasing or decreasing?
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A logistic population has carrying capacity . At approximately what population size is the population growing most rapidly?
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AP Calculus Strategy
Population-growth questions are usually testing your ability to interpret derivatives in context, not just calculate them.
- Identify exactly what the function represents.
- Determine what the derivative represents.
- Calculate or locate the derivative value.
- Use the correct units.
- Translate the numerical result into a complete sentence.
Whenever you see in a population problem, immediately ask: "How many people per unit of time is the population changing right now?"
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